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Angles in Semicircles and Chords to Tangents - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

If an angle is inscribed in a semicircle, it will be half the measure of a semicircle (180 degrees), therefore measuring 90 degrees. Angles in semicircle is one way of finding missing missing angles and lengths. Pythagorean's theorem can be used to find missing lengths (remember that the diameter is the hypotenuse). Also, the measure of an angle formed by a chord to a tangent is half the intercepted arc.

When you have an angle, inscribed inside a semicircle, there's something special that happens, so if we look at this problem right here, we have a diameter ac "but Mr. McCall how do you know that's a diameter?" Well I see that it passes through the center p and our inscribed angle we're going to call is angle abc, so notice that the vertex is on the circle and is part of a semicircle, so what do we know about semicircles? Well we know semicircles are half of a circle so the intercepted arc here between a and c is a 180 degrees because I know that half the circle is going to be over here and half the circle is going to be over here. Now, what do we know about inscribed angles? Well I know that this inscribed angle b is half of its inscribed arc and if the inscribed arc is 180 degrees then abc must be 90 degrees so no matter where you draw an inscribed arc angle in a semicircle it will always be 90 degrees because it is always having a 180 degrees as its intercepted arc and it will always be half of 180 degrees so that's one key thing about inscribed angles.
Let's look at what we know about a chord going to a point of tangent, so this is not a radius to a tangent which would be from a to d and we know that would be a right angle. What we do know is that if I draw in this chord bd its angle bde that it makes with the tangent is going to be half of its intercepted arc where the intercepted arc starts at b, it goes through c and it goes all the way down until you reach d, so this angle right here will be half of that arc bcd so two key things that you're going to use when you're trying to solve missing angles.

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